Optimal Synthesis of the Zermelo–Markov–Dubins Problem in a Constant Drift Field

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Abstract

We consider the optimal synthesis of the Zermelo–Markov–Dubins problem, that is, the problem of steering a vehicle with the kinematics of the Isaacs–Dubins car in minimum time in
the presence of a drift field. By using standard optimal control tools, we characterize the family
of control sequences that are sufficient for complete controllability and necessary for optimality
for the special case of a constant field. Furthermore, we present a semi-analytic scheme for the characterization of a (nearly) optimal synthesis of the minimum-time problem. Finally, we
establish a direct correspondence between the optimal syntheses of the Markov–Dubins and the
Zermelo–Markov–Dubins problems by means of a discontinuous
mapping.